E1-Comodules and Topological Manifolds A Dissertation presented by Anibal Medina to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Stony Brook University August 2015 ii Stony Brook University The Graduate School Anibal Medina We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Dennis Sullivan - Dissertation Advisor Mathematics Alexander Kirillov - Chairperson of Defense Mathematics John Morgan Mathematics and SCGP Gregory Brumfiel Mathematics, Stanford University This dissertation is accepted by the Graduate School Charles Taber Dean of the Graduate School iii Abstract of the Dissertation E1-Comodules and Topological Manifolds by Anibal Medina Doctor of Philosophy in Mathematics Stony Brook University 2015 The first story begins with a question of Steenrod. He asked if the product in the cohomology of a triangulated space, which is associative and graded commutative, can be induced from a cochain level product satisfying the same two properties. He answered it in the negative after identifying homological obstructions among a collection of chain maps he constructed. Using later language, his construction could be said to endow the simplicial chains with an E1-coalgebra structure. The second story also begins with a question: when is a space homotopy equivalent to a topological manifold? For dimensions greater than 4, an answer was provided by the work of Browder, Novikov, Sullivan and Wall in surgery theory, which in a later development was algebraically expressed by Ranicki as a single chain level invariant: the total surgery obstruction. After presenting the necessary parts of these stories, the goal of this work will be to express the total surgery obstruction associated to a triangulated space in terms of comodules over the E1-coalgebra structure build by Steen- rod on its chains. a los ´arboles en que estas ideas se escribieron Contents Introduction 1 1 Simplicial sets and S-coalgebras 3 1.1 Operads, coalgebras and comodules . .3 1.2 The operad S ...........................7 1.3 Simplicial sets and S-coalgebras . 16 2 Abelian sheaves and S-comodules 27 2.1 Sheaf theory of posets . 27 2.2 Simplicial complexes and Ranicki duality . 39 2.3 Topological manifolds and S-comodules . 48 A Categorical Background 57 References 65 v Introduction The goal of this dissertation is to relate the theory of algebraic surgery de- veloped predominantly by Andrew Ranicki, with that of E1-structures on chain complexes. Steenrod's construction of higher chain approximations to the diagonal inclusion has been encoded, by several authors, as a functor from simplicial sets to their normalized chains enriched with the structure of a coalgebra over an E1-operad. The first section of Chapter 1 presents the definition of an algebraic operad as well as the less common notions of coalgebra over an operad and comodule over one such coalgebra. The second section presents the specific E1-operad S related to Steenrod's construction following the work of McClure-Smith [25], Berger-Fresse [4] and others. In the last section of Chapter 1, the first of the two main technical results of this dissertation is presented as Theorem 1.3.5. It has as a corollary that the category of based ordered simplicial complexes embeds as a full subcategory into the category of S-coalgebras. Similar results have been obtained at the level of the homotopy category by Mandell [20], Smirnov [38], Smith [39] and others. The first section of Chapter 2 revisits the theory of sheaves and cosheaves over posets, see [8], [37] or [14] for other sources. It uses the connection be- tween posets and Alexandrov topological spaces, extended in Lemma 2.1.5 to a duality preserving equivalence, to emphasize the symmetry between sheaves and cosheaves over posets. This section closes with some homological algebra of such sheaves and cosheaves with values in an abelian category. In the sec- ond section of Chapter 2, the sheaf theory developed in the previous section is specialized to posets associated to ordered simplicial complexes. The notion of tensor product of functor is used to define the Ranicki duality functors of complexes of sheaves and cosheaves, whose geometry is made apparent by the pair subdivision sheaf and cosheaf. The pair subdivision sheaf is also used to define the visible symmetric complex of a regular pseudomanifold, 1 see Construction 2.2.18, which plays a central role in the application of the theory to manifold existence and uniqueness problems. The third section of Chapter 2 contains, as Theorem 2.3.4, the second main technical result of this work. It states that the category of complexes of sheaves over an ordered simplicial complex X with values in Ab embeds, as a full differential graded subcategory, into the category of comodules over the S-coalgebra C•(X). This theorem is used to relate the algebraic surgery theory of Ranicki with comodules over E1-coalgebras. In particular, Theorem 2.3.13 and Theorem 2.3.15 provide existence and uniqueness statements for ANR homology man- ifold structures and topological manifold structures on the homotopy type of a Poincar´eduality regular pseudomanifold, in terms of comodules on its S-coalgebra of chains. 2 Chapter 1 Simplicial sets and S-coalgebras Convention. The term chain complex will be reserved for a homologically graded differential graded abelian group. The category of chain complexes, 0 denoted by Ab•, is enriched over itself, i.e. HomAb• (C; C ) 2 Ab• for every 0 pair C; C 2 Ab•. In terms of this enrichment, chain maps correspond to 0-degree cycles, while chain homotopy equivalent morphisms correspond to homologous chains. 1.1 Operads, coalgebras and comodules In this section, the definition of an algebraic operad is presented as well as the less common notions of coalgebra over an operad and comodule over one such coalgebra. Definition 1.1.1. (Operad [22]) An (algebraic) operad consists of a collec- tion of chain complexes O(n); n ≥ 0, a collection of chain maps γ : O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) !O(j1 + ··· + jk); a chain map η : R !O(1) and an action of the symmetric group Σk on O(k) satisfying the following conditions. Pk O1: (Associativity) The following diagram commutes, where s=1 js = j, Pj r=1 ir = i, gs = j1 + ··· + js and hs = igs−1+1 + :: + igs for 1 ≤ s ≤ k: 3 k j j O O γ⊗id O O(k) ⊗ O(js) ⊗ O(ir) / O(j) ⊗ O(ir) s=1 r=1 r=1 γ shuffle O(i) O γ k js k O O O O(k) ⊗ O(js) ⊗ O(igs−1+q) / O(k) ⊗ O(hs) : id ⊗(⊗sγ) s=1 q=1 s=1 O2: (Unit) The following diagrams commute: =∼ =∼ O(k) ⊗ Rk / O(k) R ⊗ O(j) / O(j) 8 8 k η⊗id id ⊗ η γ γ O(k) ⊗ O(1)k; O(1) ⊗ O(j): O3: (Equivariance) The following diagrams commute, where σ 2 Σk; τs 2 Σjs , the permutation σ(j1; : : : ; jk) 2 Σj permutes k blocks of letters as σ permutes k letters, and τ1 ⊕ · · · ⊕ τk 2 Σj is the block sum: σ⊗σ−1 O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) / O(k) ⊗ O(jσ(1)) ⊗ · · · ⊗ O(jσ(k)) γ γ O(j) / O(j); σ(jσ(1);:::;jσ(k)) σ⊗σ−1 O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) / O(k) ⊗ O(jσ(1)) ⊗ · · · ⊗ O(jσ(k)) γ γ O(j) / O(j): σ(jσ(1);:::;jσ(k)) Definition 1.1.2. (Coalgebra) Let O be an operad. An O-coalgebra is a chain complex C together with chain maps θ : O(j) ⊗ C ! Cj satisfying the following conditions. 4 Pk cA1: (Associativity) Let s=1 js = j, then the following diagram commutes: γ⊗id O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) ⊗ C / O(j) ⊗ C θ id ⊗θ Cj O θk O(j ) ⊗ · · · ⊗ O(j ) ⊗ Ck / O(j ) ⊗ C ⊗ · · · ⊗ O(j ) ⊗ C: 1 k shuffle 1 k cA2: (Unit) The following diagram commutes: =∼ R ⊗ C / C : γ⊗id θ O(1) ⊗ C: cA3: (Equivariance) Let σ 2 Σj, then the following diagram commutes: σ⊗id O(j) ⊗ C / O(j) ⊗ C θ θ j j C σ / C : A morphisms of O-coalgebras is a chain map commuting strictly with all the above structure. The category of O-coalgebras will be denoted by coAlgO. Definition 1.1.3. (Comodule) Let O be an operad and C an O-coalgebra. A C-comodule is a chain complex D together with chains maps λ : O(j) ⊗ D ! D ⊗ Cj−1 satisfying the following conditions. 5 Pk cM1: (Associativity) Let s=1 js = j, then the following diagram commutes: γ⊗id O(k) ⊗ O(j1) ⊗ · · · ⊗ O(jk) ⊗ D / O(j) ⊗ M θ id ⊗λ D ⊗ Cj−1 O λ⊗θk−1 O(j ) ⊗ · · · ⊗ O(j ) ⊗ D ⊗ Ck−1 / O(j ) ⊗ D ⊗ · · · ⊗ O(j ) ⊗ C: 1 k shuffle 1 k cM2: (Unit) The following diagram commutes: =∼ R ⊗ D / D : γ⊗id θ O(1) ⊗ D: cM3: (Equivariance) Let σ 2 Σj−1 ⊂ Σj, then the following diagram com- mutes: σ⊗id O(j) ⊗ D / O(j) ⊗ D θ θ D ⊗ Cj−1 / D ⊗ Cj−1: id ⊗σ A morphisms of C-comodules is a homeomorphism of abelian groups commuting strictly with all the above structure. The category of O-comodules O is enriched over Ab• and will be denoted by coModC . Example 1.1.4. The operad A has A(j) = Z[Σj] with unit map equal to the identity and product maps dictated by the equivariance formulas. An A-coalgebra C is the same thing as a coassociative coalgebra.